Special Families of Sets and Baer-Specker Groups

ABSTRACT

We prove that the Baer-Specker group Π =  contains a pure subgroup isomorphic to the direct sum of 2 _⁰ copies of itself. We produce 2 ^{ℵ ₀} nonisomorphic subgroups of Π, each isomorphic to its dual group. Finally, we show that the isomorphism type of a generalized product of ℤ's, the set of functions I → ℤ with support of size at most α, uniquely determines both the cardinality of I and α (as long as there are no measurable cardinals ≤α). All three of these results are obtained using set-theoretic existence theorems, namely, the existence of large independent families, large almost disjoint families, and Δ-systems.

Key Words:

• Almost disjoint sets
• Baer-Specker group
• Delta systems
• Dual groups
• Independent sets

Mathematics Subject Classification:

• Primary 03E75
• 20K25
• Secondary 03E05
• 20K20

ACKNOWLEDGMENT

The first author was partially supported by NSF grant DMS–0070723.

Notes

^#Communicated by K. Rangaswamy.